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G. Dlubek et al.: o-Ps Lifetime Distribution in Polymers (II) 303
phys. stat. sol. (a) 172, 303 (1999)
Subject classification: 61.41.+e; 71.60.+z; 78.70.Bj; S12
Do MELT or CONTIN Programs Accurately Reveal
the o-Ps Lifetime Distribution in Polymers?
II. Semicrystalline Polymers 1 †
G. Dlubek (a), Ch. HuÈbner (b), and S. Eichler (b)
(a) ITA Institut fuÈ r innovative Technologien GmbH, KoÈthen, Auûenstelle Halle,
Edvard-Grieg- Weg 8, D-06124 Halle/Saale, Germany
e-mail: gdlubek@aol.com
(b) Fachbereich Physik, Martin-Luther-UniversitaÈt Halle-Wittenberg,
D-06099 Halle/Saale, Germany
(Received November 12, 1998; in revised form February 2, 1999)
Positron lifetime experiments of high statistical accuracy (50 M total coincidence count) were performed
for various polyethylene samples with different degrees of crystallinity of X c = 43, 60, and
70%. Four lifetime components were resolved using the MELT, CONTIN, or the discrete-term
LIFSPECFIT routines. Two long-lived components appear, their lifetimes (t 3 = 1.1 to 1.2 ns and
t4 = 2.6 to 2.7 ns) and intensities (I3 = 5 to 8% and I4 = 13 to 24%) vary systematically with Xc.
t4 arises from o-Ps annihilation in holes of the amorphous phase while t3 may be attributed to
crystals. From a detailed analysis of computer-generated spectra it was concluded that artefacts
may appear in the analysed lifetime parameters t i and Ii when the fourth lifetime is assumed to
have a (Gaussian) distribution (due to a hole size and shape distribution). The artefacts are due to
the inability of the routines correctly to mirror very broad lifetime distributions. The most significant
effect is an increase in the analysed parameters t 3, t4 and I3 but a decrease in I4, compared
with the simulation input, with increasing width of the simulated distribution of the fourth lifetime.
From this it follows that care must be taken when discussing values of and changes in the lifetime
parameters of semicrystalline or other two-phase polymers. A comparison of the parameters analysed
from experimental spectra with those from simulated ones appears always advisable.
Positronenlebensdauerexperimente mit hoher statistischer Genauigkeit (Gesamtzahl von Koinzidenzereignissen
50 Millionen) wurden fuÈ r verschiedene Polyethylenproben mit unterschiedlichem
KristallinitaÈtsgrad von X c = 43, 60 und 70% durchgefuÈ hrt. Unter Nutzung der Programme MELT,
CONTIN oder LIFSPECFIT (diskrete Zerfallsterme) wurden vier Lebensdauerkomponenten aufgeloÈ
st. Es treten zwei langlebige Komponenten auf, deren Lebenszeiten (t 3 = 1,1 ±1,2 ns und t4 =
2,6 ±2,7 ns) und IntensitaÈten (I3 =5±8% und I4 =13±24%) systematisch mit Xc variieren. Aus
einer detaillierten Analyse von computergenerierten Spektren wurde geschluûfolgert, daû Artefakte
in den analysierten Lebensdauerparametern ti und Ii auftreten koÈ nnen, wenn vorausgesetzt
wird, daû die vierte Lebenszeit eine (Gauss-foÈ rmige) Verteilung aufweist (die auf eine HohlraumgroÈ
ûen- und Formverteilung zuruÈ ckzufuÈ hren ist). Die Artefakte treten aufgrund der mangelnden
Eignung der Auswerteprogramme auf, sehr breite Lebensdauerverteilungen genau zu widerzuspiegeln.
Der bedeutendste Effekt ist ein Anstieg in den analysierten Parametern t 3, t4 und I3 und ein
Abfall in I4, verglichen mit den Ausgangsparametern der Simulation, mit wachsender Breite der
simulierten Verteilung der vierten Lebenszeit. Daraus folgt, daû bei der Diskussion von GroÈ ûen
und Ønderungen in den Lebensdauerparametern von teilkristallinen oder anderen zweiphasigen
Polymeren Vorsicht geboten ist. Ein Vergleich von Parametern, die aus experimentellen Spektren
analysiert werden, mit jenen aus Simulationen erscheint in jedem Fall ratsam.
1 † Part I see phys. stat. sol. (a) 168, 333 (1998).

304 G. Dlubek, Ch. HuÈbner, and S. Eichler
1. Introduction
Positron annihilation lifetime spectroscopy is widely used to investigate subnanometer
size holes [1 to 4] which appear in amorphous polymers as a consequence of the irregular
arrangement of molecular chains and form the so-called free volume, see e.g. [5]. In
polymers, a fraction of the positrons entering the material form positronium, the lifetime
of the ortho state of this (o-Ps) [6] decreases from 142 ns in vacuum, typically to
the low ns-range in matter, due to annihilation during collisions of Ps with molecules
(pick-off annihilation). The o-Ps pick-off lifetime very sensitively reflects the size of the
local free volumes in which Ps is confined [2, 7, 8].
In previous works of our group [9, 10], special attention was paid to the distribution
of o-Ps lifetimes analysed with the routines CONTIN [11] and MELT [12]. It is widely
assumed that this distribution in lifetimes is due to the annihilation of o-Ps in freevolume
holes that have a distribution of sizes (and shapes) [2, 13, 14]. We, and also
other groups [15, 16], observed that the width of the analysed distribution shows a
rather complicated behaviour as function of the regularisation in the analysis routines,
the real width of the distribution (assumed in simulations), and the total number of
counts in a spectrum. For broad distributions the analysis routines deliver two apparent
sub-lifetimes centred approximately symmetrically to their common mass centre. The
appearance of such sub-lifetimes ( 1.7 and 2.4 ns) we have actually observed in lifetime
spectra of amorphous polycarbonate and polystyrene measured with very high
statistical accuracy [9]. We found that it is not possible to distinguish whether the two
analysed near-neighbour lifetimes appear as an artefact of the analysis of a broad o-Ps
lifetime distribution, or are due to two actually existing (discrete or narrowly distributed)
neighbour o-Ps lifetimes. These conclusions lead to a critical view of the current
use of the routines in estimating o-Ps lifetime distributions, and from these the freevolume
hole size distributions in polymers.
This paper continues our previous studies, and discusses a more complicated situation. In
some polymers, or other molecular materials, the appearance of two well separated o-Ps
lifetimes has been observed and attributed to two different o-Ps states. In polytetrafluoroethylene
(PTFE, [11, 18, 19]) and in polyethylene (PE, [19 to 22]), for example, an o-Ps
lifetime of 1.0 to 1.5 ns appears in addition to the 4 ns (3 ns) component. The latter (larger)
one is due to holes in the amorphous phase. The nature of the medium o-Ps component is
not entirely clear and attributed to o-Ps annihilation from the (densely packed) crystals of
the semicrystalline polymers [19, 20, 22], or to a state attached to molecules [18, 21].
In this paper, we attempt to address the question of whether the analysis of lifetime
spectra provides realistic lifetimes and their intensities in the case where two different
o-Ps states exist, and the longer-lived ones exhibit a lifetime distribution. This situation
can be considered to be typical for semicrystalline or other two-phase polymers. For
this study, positron lifetime spectra of PE of different crystallinity were measured with
a very high statistical accuracy of 50 M counts below each spectrum. The spectra were
analysed both in terms of discrete components and continuous distributions. The reliability
of our results was proved by analysing computer-generated spectra and comparing
the results with the experimental ones. We focus our discussion on results obtained
with the routine MELT (and the discrete-term routine LIFSPECFIT [23]) because we
found that MELT responses similarly but more sensitively to near neighbour and also
distributed lifetimes than CONTIN does [9, 10].

Do MELT or CONTIN Programs Accurately Reveal o-Ps Lifetime Distribution? (II) 305
2. Experimental Investigations
2.1 Experimental
For the positron lifetime experiments, a fast±fast coincidence system (see for example
[6]) with a time resolution (full width at half maximum, FWHM, of a Gaussian resolution
function) of 235 ps and a channel width of 50.16 ps was used. The background in
the spectra was 32 per 1 M of total coincidence count.
The specimens were platelets of the commercial polymers of high density polyethylene
(HDPE1 and 2, crystallinity Xc = 70 and 60%) and low-density polyethylene
(LDPE, crystallinity Xc = 43%), and of 8 8 mm 2 area and 1.5 mm thickness. The crystallinity
of PEs was estimated from differential scanning calorimetry (DSC) experiments.
For the lifetime studies, two identical samples were sandwiched around a
6.5 10 5 Bq positron source, prepared by evaporating carrier-free 22 NaCl solution on
an aluminium foil of 2 mm thickness. For each specimen, a series of five measurements
was performed, each collecting 10 M total counts in a single spectrum. Following inspection
of the time zero, the 10 M count spectra were added to give a final spectrum
of 50 M coincidence counts. Before and after the series of measurements, reference
spectra (well annealed aluminium platelets, t = 162 ps) containing a total of 20 M
counts were measured, and from these the width of the resolution function and the
lifetime components due to annihilation in the positron source ( 22 NaCl: 380 ps/1.7%; Al
cover: 180 ps/2.0%; 2500 ps/0.5% surface effects) were estimated.
2.2 Data analysis
The positron lifetime spectrum is conventionally described by a sum of discrete exponentials
s…t† ˆ P …Ii=ti† exp ‰ …t=ti†Š ; …1†
each having a characteristic positron lifetime of ti with a relative intensity of Ii, P Ii = 1
[6]. Assuming that the positron lifetimes t follow a distribution, the lifetime spectrum
may be described by a continuous decay form
s…t† ˆ „1
I…t† …1=t† exp … t=t† dt ; …2†
0
where „ I(t)dt = 1 [2, 11, 12]. The observed spectrum may be expressed by
y…t† ˆ R…t†* …Nts…t† ‡ B† ; …3†
where * denotes the convolution of the decay integral with the resolution function R(t),
and B and Nt are the background and the total count of the spectrum, respectively.
The MELT [12] routine inverts the lifetime spectrum into a continuous lifetime distribution
using a quantified maximum entropy method. The input parameters of this routine
are the lifetimes and intensities due to annihilation in the positron source, and the
time zero channel of the spectrum. In the discrete-term LIFSPECFIT [23] analysis the
model function (1) together with (3) is least-squares fitted to the experimental data
points. The number of exponentials, the FWHM of the resolution function and the
source corrections are needed as input parameters. The analysis of the experimental
results employing the two routines were carried out with the same parameters mentioned
in the preceding papers [9, 10].

306 G. Dlubek, Ch. HuÈbner, and S. Eichler
2.3 Results of experiments
Results of the MELT analysis of experimental positron lifetime spectra are shown in
Fig. 1 and Table 1. The MELT routine employs a procedure to penalise deviations of
the solutions from smoothness. The smoothing is controlled by the entropy weight E.
For small E, no solution may be found due to singularities in the calculated matrix. For
large E the solution may be oversmoothed. A preferred solution may be chosen on the
basis of the variance and of some other parameters. The distributions are calculated for
an entropy weight of E = 5 10 ±8 which was chosen to give the ``preferredº solution
based on the criterion given by Shukla et al. [12]. The variance of the analysis was
1.2. Four well separated peaks can be observed, their characteristic lifetimes ti and
intensities Ii were calculated as the mass centre of and the area below the i-th peak.
The four lifetimes (Table 1) are well known [16 to 22] and commonly attributed to p-Ps
self-annihilation (t1 170 ps) free positron annihilation (t2 380 ps) and o-Ps pick-off
annihilation (t3 = 1.1 to 1.2 ns and t4 = 2.6 to 2.7 ns). t3 and t4 are spaced by a factor
of 2.3 to 2.4. This factor is higher than the theoretical resolution factor of the experiment
(2.0, estimated following Ref. [24]). In the analysis of computer-generated spectra
we observed that MELT is able to resolve lifetimes spaced by a factor of only 1.5 from
50 M total count spectra [9, 10]. Therefore we can assume that both lifetimes mirror
the appearance of two physically different Ps states. The smaller o-Ps lifetime of
t3 = 1.10 to 1.20 ns may be attributed to o-Ps annihilation from crystals [19, 29, 22], the
larger one of t4 = 2.60 to 2.70 ns from holes in the amorphous phase [17 to 22].
Unfortunately, the analysed medium lifetime t3 is rather unstable and its value is
largely affected by a correct estimation of the resolution function and of the time zero
to the spectrum. In literature, t3 lifetimes between 0.8 and 1.6 ns may be found [19 to
22]. As can be observed in Fig. 1 (and Table 1) t3 and t4 increase with decreasing
crystallinity of PE. Parallel to this, I4 also shows a distinct increase while I3 shows a
Fig. 1. Positron lifetime distribution in polyethylene samples of different crystallinity Xc analysed
with the MELT routine. Filled circles: HDPE1, Xc = 70%; empty circles: HDPE2, Xc = 60%; open
triangles: LDPE, Xc = 43%. (50 M total count, entropy weight E = 5 10 ±8 )

Do MELT or CONTIN Programs Accurately Reveal o-Ps Lifetime Distribution? (II) 307
Ta b l e 1
Positron lifetime parameters in polyethylene samples of different crystallinity Xc estimated
via the routines MELT and LIFSPECFIT (four discrete terms are assumed).
Polymer Xc (%) routine t1 (ps) t2 (ps) t3 (ps) t4 (ps)
HDPE1 70 MELT 170 373 1113 2572
LIFSPECFIT 174 3 379 5 1160 60 2603 32
HDPE2 60 MELT 171 372 1140 2622
LIFSPECFIT 181 3 384 7 1152 60 2643 22
LDPE 43 MELT 182 382 1250 2680
LIFSPECFIT 192 3 394 9 1236 60 2702 26
Polymer Xc (%) routine I1 (%) I2 (%) I3 (%) I4 (%)
HDPE1 70 MELT 23.3 3 56 3 7.6 1.5 13.1 1.5
LIFSPECFIT 25.5 1.1 53.3 0.8 8.5 0.4 12.7 0.5
HDPE2 60 MELT 25.3 3 50 3 8.2 1.5 16.5 1.5
LIFSPECFIT 26.2 1 48.9 0.8 9.3 0.4 15.5 0.4
LDPE 43 MELT 25.8 4 45.7 4 4.9 1.6 23.6 1.1
LIFSPECFIT 33.2 0.6 37 0.9 7.2 0.3 22.6 0.5
slight and not very definite decreases. At this point we notice that the lifetimes obtained
from LIFSPECFIT are generally somewhat larger than those from MELT (and
CONTIN). Moreover, the intensity ratio I1/(I3 + I4) = 0.9 to 1.1 is distinctly larger than
the theoretical expectation of Ip-Ps/Io-Ps = 0.33 [6]. This effect is frequently observed in
the literature and commonly attributed to the limited resolution of the lifetime technique
for small lifetimes. This problem is beyond the scope of our current work, but we
will discuss it in large detail in a subsequent paper.
3. Analysis of Computer-Generated Spectra
3.1 Simulation of decay curves
The computer-generated lifetime spectra are assumed to contain four different components,
which originate from the annihilation of p-Ps (t1), free (not Ps) positrons (t2),
and o-Ps in crystalline (t3) and amorphous regions (t4). It was assumed that each of the
lifetime components exhibits a Gaussian distribution,
I…t† ˆ P Ii…1=si…2p† 1=2 † exp ‰ …t ti† 2 =2s 2 i †Š : …4†
The standard deviation s4 of the upper o-Ps Gaussian was varied between 0 and 800 ps
in steps of 100 ps. Fixed narrow lifetime distributions were assumed for the t1, t2, and
t3 components. The four-component lifetime spectra were simulated assuming the following
parameters chosen to represent the experimental situation in semicrystalline
(low density) PE:
t1 = 170 ps; s1 = 5 ps; I1 = 0.20,
t2 = 375 ps; s2 = 20 ps; I2 = 0.50,
t3 = 0.9 ns; s3 = 60 ps; I3 = 0.06,
t4 = 2.5 ns; s4 = 0; 100; . . . 800 ps I4 = 0.24.

308 G. Dlubek, Ch. HuÈbner, and S. Eichler
Furthermore, lifetime spectra with different t 4 and I 4 (I 3) were also generated. Spectra
with total counts of 10 M, 50 M and 300 M were simulated (for more details of simulation
see Refs. [9, 10]). A background of 30 counts per 1 M total count and appropriate
Poisson noise were added to each channel. Further parameters of the simulations were:
channel width: 50 ps; total number of channels: 1000, time zero channel: 50; FWHM of
Gaussian resolution function R(t): 235 ps. To take into account annihilation events from
the positron source itself, three components with the same lifetimes and intensities as
observed in the experiments were included in the final spectrum.
3.2 Results of simulation
In Fig. 2 the results of the MELT analysis of two computer-generated lifetime spectra
containing 300 M coincidence counts (background B = 0, filled circles) and 50 M counts
(B = 1500, empty circles) together with the o-Ps lifetime distribution assumed in simulation
(s 3(SIM) = 60 ps, s 4(SIM) = 500 ps) are shown. The mass centre and the intensity of
the smaller two lifetimes t1 ( 170 ps) and t2 ( 375 ps) agree well with the simulation
input except that the distributions are distinctly smoothed. In the range of t = 0.8 to
4 ns, three lifetime peaks appear in the distribution of the 300 M count/B = 0 spectrum.
The smaller lifetime, t3 = 0.93 ns, is somewhat larger than the simulation input, the
same is observed for I3 = 6.2%. The two larger lifetimes originate apparently from the
component t4(SIM) = 2.5 ns/s4(SIM) = 500 ps/I4(SIM) = 24%. The artefact of the splitting
of broad distributions analysed via MELT (or CONTIN) into two sub peaks situated at
approximately t 4(I) = t 4(SIM) ± s 4(SIM) and t 4(II) = t 4(SIM) + s 4(SIM) with intensities of
I4(I) I4(II) I4(SIM)/2 was observed and discussed in detail in our previous papers for
the case of amorphous polymers [9, 10]. From the parameters of these sub peaks,
t4(I) = 1.98 ns/I4(I) = 11.7% and t4(II) = 2.96 ns/I4(II) = 11.9%, mean parameters of
Fig. 2. Positron lifetime distribution analysed from simulated lifetime spectra with the MELT routine.
Filled circles: 300 M total count, no background, entropy weight E = 1 10 ±8 ; empty circles:
50 M total count, background B = 1500, E = 5 10 ±8 ; curve without symbols: o-Ps lifetime distribution
used as simulation input

Do MELT or CONTIN Programs Accurately Reveal o-Ps Lifetime Distribution? (II) 309
(t 4(I) I 4(I) + t 4(II) I 4(II) )/(I 4(I) + I 4(II)) = 2.47 ns and I 4(I) + I 4(II) = 23.6% follow which
correspond well to the values of t4 and I4 assumed in simulations.
Only two o-Ps lifetime peaks appear in the distribution of the 50 M count/B = 1500
spectrum. Both, t3 and t4 are distinctly larger than assumed in simulations. These
shifts appear to be due to the inability of the MELT routine (the same is true for
CONTIN) correctly to mirror very broad lifetime distributions. The analysed components
t3 = 1.08 ns/I3 = 10.1% and t4 = 2.65 ns/I4 = 19.3% can be considered as
coming from a mixing of the lower o-Ps lifetime (t3(SIM)) and parts of the medium
sub-lifetime (t 4(I)), and parts of the medium (t 4(I)) and the upper sub-lifetimes (t 4(II))
observed in the distribution analysed from the 300 M count/B = 0 spectrum (Fig. 2).
The reason for this behaviour is the lower statistical accuracy of the 50 M count
spectrum and the statistical scatter due to the background. This leads to a loss in the
information content of spectra which is important for the analysis of the distributed
t4 lifetimes (see also the discussion in Ref. [24]). Similar effects appear if a too low
number of data points is used in the spectrum analysis. In a semi logarithmic plot,
Fig. 3. Characteristic lifetimes t3 and t4 and their intensities I3 and I4 analysed with the help of the
MELT (E = 5 10 ±8 , empty symbols) and the discrete-term routine LIFSPECFIT (filled symbols)
from computer-generated spectra (total count of 50 M) vs. s4(SIM). The solid lines correspond to
the simulated lifetime parameters. The dash-dotted lines indicate t4(I) = t4(SIM) ± s4(SIM) and t4(II)
= t4(SIM) + s4(SIM) and the dotted lines represent [I3(SIM) + I4(SIM)/2] = 18% and I4(SIM)/2 = 12%

310 G. Dlubek, Ch. HuÈbner, and S. Eichler
discrete lifetime components have a slope of ±1/t i. For s 4 = 500 ps the behaviour of
log y(t) is curved: in the range t < 8 ns (t > 8 ns) y(t) lies below (above) the spectrum
for s4 = 0 ps. For spectra with low total count or high background the tail of
the distribution at high lifetimes cannot be analysed correctly for that reason. The
background, due to random coincidence events, may be reduced (in practice by a
factor of approximately 3) by using weak positron sources. This, however, is not sufficient
to improve the sensitivity of the analysis significantly.
In the following we discuss the analysed lifetimes and intensities of the o-Ps components
as a function of parameters s 4(SIM), t 4(SIM) and I 4(SIM). In Fig. 3 the behaviour of
the analysed lifetimes and intensities of the third and fourth lifetime component (analysed
from 50 M/B = 1500 spectra) as function of s4(SIM) is shown. With increasing
s4(SIM) the analysed lifetime t4 behaves approximately between t4(SIM) and t4(I) =
t4(SIM) + s4(SIM), and t3 shows also a similar increase. Simultaneously with the increase
in the lifetimes, the intensity I3 increases while I4 decreases. These effects appear more
pronounced in the discrete-term (LIFSPECFIT) analysis than in the MELT results. The
reason for this is that MELT responds to some extent to the distribution in the fourth
lifetime while the discrete-term routine LIFSPECFIT does not. In the limit t 3(SIM) =
t4(SIM) ± s4(SIM) we may expect in a simple picture t3 t3(SIM) = t4(SIM) ± s4(SIM), t4
t4(SIM) + s4(SIM), I3 I3(SIM) + I4(SIM)/2 = 18% and I4 I4(SIM)/2 = 12%. Due to the
Fig. 4. As in Fig. 3, but lifetimes and their intensities vs. t4(SIM). s4(SIM) was fixed to 500 ps

Do MELT or CONTIN Programs Accurately Reveal o-Ps Lifetime Distribution? (II) 311
background effect discussed previously, t 4 may behave below the limit t 4(I) = t 4(SIM) +
s4(SIM). The trend in the lifetime parameters analysed from spectra of different total
count of 300 M, 50 M and 10 M is the same. Only the scatter in the parameters increases
with decreasing total count.
The behaviour of the analysed lifetimes and intensities of the third and fourth lifetime
component as function of the simulated lifetime t4(SIM) is shown in Fig. 4 (s4(SIM)
= 500 ps, I4(SIM) = 24%, t3(SIM) = 900 ps, and I3(SIM) = 6%). For t4(SIM) = 1400 ps
values of t3 = 1147 ps and t4 = 2008 ps are obtained, which are higher than t3(SIM) =
t 4(SIM) ± s 4(SIM) = 900 ps and t 4(SIM) + s 4(SIM) = 1900 ps. With increasing fourth lifetime
the overestimation of t4(SIM) in the analysed t4 becomes smaller. t3 increases to a
maximum value of 1400 ps and approaches t3(SIM) = 900 ps for large t4(SIM). For small
t4(SIM) the intensity I3 (I4) is distinctly higher (smaller) than assumed in simulations, while
for large t4(SIM) the analysed intensities approach the simulated ones. Only I3 is somewhat
lower (and t3 somewhat higher) than simulated. This is because, due to a slight overestimation
in the values of t1 and t2, their intensities I1 and I2 include some fraction of the
higher ones. Similar effects were already observed and discussed previously [9, 10, 25].
In Fig. 5 the analysed parameters are plotted as function of the intensity of the
fourth lifetime component I4(SIM), where I3(SIM) + I4(SIM) = 30% has been assumed
(s4(SIM) = 500 ps, t3(SIM) = 900 ps, and t4(SIM) = 2500 ps). As discussed previously, t3
Fig. 5. As in Fig. 3 but lifetimes and their intensities vs. I4(SIM). I4(SIM) + I3(SIM) was fixed to 30%, and
s4(SIM) to 500 ps. The dotted lines indicate [I3(SIM) + I4(SIM)/2] and I4(SIM)/2 while the dashed line
corresponds to the average lifetime [t3(SIM) I3(SIM) + (t4(SIM) ± s4(SIM)) (I4(SIM)/2)]/[I3(SIM) + I4(SIM)/2]

312 G. Dlubek, Ch. HuÈbner, and S. Eichler
behaves as being due to a mixture of t3(SIM) and a part of the t4(I) = t4(SIM) ± s4(SIM) component, while t4 seems to be a mixture of the t4(II) = t4(SIM) + s4(SIM) and a part of
the t4(I) = t4(SIM) ± s4(SIM) component but dominated by t4(II). For small I4(SIM) values,
t3 is dominated by t3(SIM), but for larger I4(SIM) by t4(I) = t4(SIM) ± s4(SIM). For I4(SIM)
= 30%, I3(SIM) = 0, lifetimes of t3 = t4(I) = t4(SIM) ± s4(SIM) and t4 = t4(II) = t4(SIM) +
s4(SIM) are analysed.
For small I4(SIM), the value of I3 closely matches I3(SIM), while I4 is smaller than
I4(SIM). Again the trend I3 + I4 < I3(SIM) + I4(SIM) is observed. Above I4(SIM) 22% an
inflection (reversal) in the behaviour of I3 and I4 can be seen. I3 and I4 separate clearly
from I3(SIM) and I4(SIM) and both approach I4(SIM)/2 = 15%. That is the range where t3
and t4 approach t4(I) = t4(SIM) ± s4(SIM) and t4(II) = t4(SIM) + s4(SIM).
From the plots in the Figs. 3 to 5 we can conclude that in the case of distributed o-Ps
lifetimes the analysed lifetime parameters t3, t4, I3, and I4 may deviate considerably
from the real ones. All of these analysed parameters may vary if only one of the real
(or in our case simulated) parameters changes: s4(SIM), t4(SIM) (or t3(SIM)) and I4(SIM)
(or I3(SIM)). 4. Discussion
In the experimental results for PE collected in the Fig. 1 and Table 1 we observe the
following trends with decreasing crystallinity: (i) increase in t4, (ii) increase in I4, (iii)
increase in t3, and (iv) a possible but slight decrease in I3. The third and fourth lifetime
components were previously attributed to o-Ps annihilation from crystals and from
holes in the amorphous phase of the semicrystalline polymers. The holes have a size
and a shape distribution which results in a lifetime distribution in the fourth component.
o-Ps lifetime distributions with standard deviations of 350 to 400 ps have been
previously analysed for glassy polycarbonate (PC) and polysterene (PS) [9]. Since our
measuring temperature (25 C) is well above the glass transition of PE (Tg = ±78 C
[19]) the average hole size is larger than in PC (Tg = 150 C) and PS (Tg = 100 C)
which exhibit o-Ps lifetimes of 2 ns [9]. From this we may expect that the o-Ps lifetime
distribution in PE to be approximately the same or probably broader than in PC
and PS. A broad (t 4) o-Ps lifetime distribution may cause artefacts in the experimental
parameters as discussed in the earlier section.
The observed t4 of 2572 ps for PE of Xc = 70% crystallinity corresponds to a mean
size of the free-volume holes confining o-Ps of radius r = 0.333 nm and volume v =
0.155 nm 3 when applying the well known and accepted semiempirical relation between
t4 and r due to Eldrup et al. [7], and Nakanishi and Jean [8, 2]. From Fig. 3 it follows,
however, that the analysed t4 may overestimate the true one, that is the mass centre
of the corresponding o-Ps lifetime distribution. The degree of overestimation is highly
controlled by the value of s 4 (Fig. 3) but depends also on t 4 itself (Fig. 4) and on I 4
(Fig. 5). The parameters t3/I3 = 1113 ps/7.6% and t4/I4 = 2572 ps/13.1% analysed
from the experimental lifetime spectra of PE having 70% of crystallinity (Table 1),
for example, may come from original parameters of approximately 800 ps/6% and
2400 ps/17% assuming s4 = 500 ps.
The increase in the observed t4 from 2572 ps for PE of Xc = 70% crystallinity to
2680 ps for Xc = 43% (Fig. 1, Table 1) correspond to an increase in the mean size of the freevolume
holes from r = 0.333 nm (v = 0.155 nm 3 ) to r = 0.341 nm (v = 0.166 nm 3 ).

Do MELT or CONTIN Programs Accurately Reveal o-Ps Lifetime Distribution? (II) 313
Such an increase has been observed already in Ref. [20] and is physically reasonable,
not directly due to the crystallinity, but due to the branching of PE. The ethylene
chains in HDPE (70 and 60% crystallinity) are almost linear and have only a small
number of branches ( 10 methyl groups per 1000 CH2 units [26]). The alignment of
chains is little disturbed, therefore, the crystals can easily be formed and the holes in
the amorphous phase are probably small and very likely to be highly anisotropic. The
anisotropy reduces t4 with respect to three-dimensional holes having the same volume
[27]. The holes are expected to get larger and become more three-dimensional the more
PE is branched due to sterical hindrances caused by the branches. Due to the same reason
the crystallinity decreases with increasing branching. The number of branches in LDPE
(43% crystallinity) is typically 50 methyl groups per 1000 CH2 units [26]).
Compared with the true parameters, the analysed t4 may include an extra increase
due to a possible, with the hole size increasing, broadening of the hole volume (and
eventually hole shape) distribution (Fig. 3). Furthermore, an increase in I4 may also
cause some additional increase in t4 (Fig. 5). An increasing separation between t4 and
t 3, on the other hand, reduce the overestimation of t 4 (Fig. 4).
The lifetime t 3 seems to be mostly affected by the artefacts. Lightbody et al. [28]
have observed that the o-Ps lifetime to-Ps in molecular crystals decreases with increasing
packing coefficient C. The data of Ref. [28] may be fitted by a straight line: to-Ps =
7.92 ±9.616 C (see also Ref. [19]). From this relation an o-Ps lifetime of to-Ps = 900 ps
would be expected for PE crystals. This is the value assumed in simulations. The packing
coefficient may be estimated via C = 4VW/(a b c) = 0.017/0.0233 = 0.73 where
VW = 0.017 nm 3 is the van der Waals volume [29] and (a b c)/4 = 0.0233 nm 3 the
total volume per CH 2 group at room temperature (a, b and c denote the lattice parameters
of the orthorhombic unit cell of PE [26]). The observed t3 values of 1100 to
1200 ps are clearly larger than 900 ps. This could have physical reasons and may be
due, to some extent, to an averageing over the o-Ps lifetimes coming from ``perfectº
crystalline regions and from ``relaxedº crystal defects [30]. Non-relaxed defects are expected
to exhibit o-Ps lifetimes like holes in the amorphous phase and are, therefore,
included into the fourth lifetime component.
As can be concluded from Figs. 3 to 5, high t3 values could be also an artefact caused by
the lifetime distribution in the fourth component. Owing to our simulations the increase
in t3 with decreasing crystallinity (Table 1) could be an artefact of the spectrum analysis
due mainly to the increase of I4 (Fig. 5) and to a possible increase of s4 (Fig. 3).
The o-Ps intensity I4 reflects the relative probability of o-Ps annihilations from holes
in the amorphous phase. Its increase with increasing fraction of the amorphous phase
(decreasing Xc) is therefore easy to understand. Moreover, larger holes may also promote
the Ps formation [31]. The analysed I4 is underestimated for larger widths s4 of
the o-Ps distribution (Fig. 3). This underestimation reduces with increasing separation
between t 4 and t 3 (Fig. 4).
The intensity I3 reflects the relative probability of o-Ps annihilations from crystals. Its
analysed value is clearly lower than expected from the volume fraction of the crystalline
phase Xc, although it should be overestimated as an artefact of the lifetime distribution
in t4 (Fig. 3). Some physically reasonable arguments may be found for the low
I3. n-alkanes have the same crystalline structure as polyethylene crystals; the number of
carbon atoms in one chain is much smaller (20 to 30 in Ref. [32]) and also the packing
coefficient is smaller (0.67 [32]) than in PE crystals. In more or less perfect crystals of

314 G. Dlubek, Ch. HuÈbner, and S. Eichler
various n-alkanes an o-Ps component of 1.60 ns/15% was observed, which changed
due to the formation of conformational (open volume) defects near the melting point to
3.00 ns/30% [32]. From this we may conclude on a ratio of the Ps formation probability
in crystals and in the amorphous phase in PE of Pc /Pa 1/2. Assuming Xc = 43% (70%)
we estimate I3 /I4 = PcXc/[Pa(1 ± Xc)] = 0.33 (1.17). This is clearly higher than the observed
I3 /I4 = 4.9%/23.6% (7.6%/13.1%) = 0.21 (0.58). The discrepancy may be attributed
to a lower Ps yield Pc in the more densely packed PE crystals compared with crystalline
n-alkanes. The discrepancy also could be due to an out-diffusion of o-Ps from the PE
crystallites and/or their trapping and annihilation at unrelaxed, hole-type defects inside
the crystals. Both effects would result in a decrease in I3 but in an increase in I4. The Ps
diffusion coefficient in polymer crystals is not known, but for crystals of ice a value of DPs
= 0.2 10 ±4 m 2 s ±1 [33] has been estimated from which a Ps diffusion length of Lo-Ps =
(6 DPs to-Ps) 1/2 = 350 nm follows. For amorphous polymers DPs = (2.6 to 3.2) 10 ±10 m 2
s ±1 and Lo-Ps 2 nm has been estimated [34]. Since the typical thickness of crystalline
lamellae in PE is in the range of 10 nm [26], we can expect that a large fraction of o-Ps
formed inside the crystals migrates out of them and annihilates in the amorphous phase
(or in the amorphous-like interface between crystallites and amorphous regions).
We expect a decrease of I3 with decreasing crystallinity which could be enhanced due to
the increase in t4 (Fig. 4). An increase in s4, on the other hand, may compensate this at
least partially (Fig. 3). For small original I3 both analysed o-Ps intensities, I3 and I4, may
exhibit a complicated behaviour and the analysed I3 may possibly show an increase with
decreasing original I3 (Fig. 5). The integral influence of the various artefacts discussed on
the analysed parameters can be expected to be responsible for the observation that the
decrease in the analysed I 3 with decreasing X c is not as clear as expected.
5. Conclusion
The results of our experiments show that four lifetimes appear in the lifetime spectra of
PE, the long-lived components may be attributed to o-Ps annihilation from crystals (t3)
and from holes in the amorphous phase (t 4). The lifetimes and corresponding intensities
vary systematically and in a physically reasonable way with the degree of crystallinity
(branching) of the PEs. As found, however, from the analysis of computer-generated
spectra, the values of analysed lifetimes and intensities may be largely affected by
artefacts appearing in the analysis of distributed lifetimes (t4). The artefacts are due to
the inability of the routines correctly to mirror very broad lifetime distributions. There
is a complicated and mutual influence of original (simulated) lifetimes, intensities and
especially of the widths in the distribution of the longer o-Ps lifetime on the analysed
parameters. The most significant effect is an increase in the analysed t 3, t 4 and I 3 but a
decrease in I4, compared with the simulation input, with increasing width of the simulated
distribution of the fourth lifetime. From this it follows that care must be taken
when discussing values of and changes in the lifetime parameters of semicrystalline or
other two-phase polymers. A comparison of the parameters analysed from experimental
spectra with those from simulated ones appears always advisable.
Acknowledgements The authors wish to thank A. Shukla (Geneva) and P. HautojaÈrvi
(Helsinki) for supplying the PC versions of the routines MELT and LIFSPECFIT. H.-J.
Radusch (Merseburg) is acknowledged for supplying the PE samples. D. Bamford and
I. Wilkinson (Bristol) are acknowledged for critical reading of the manuscript.

Do MELT or CONTIN Programs Accurately Reveal o-Ps Lifetime Distribution? (II) 315
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